Its very weird, normally it would be OK, as bazzite ships X11 with special Nvidia drivers so it has no reason to bug out as your GPU is maybe the most common gpu of all time according to steam hardware survey so driver support should not be a problem. I would make sure that the GPU is fully working especially if you got it recently from second hand market, and that the cable from the GPU enclosure to your laptop is rated with proper speeds and also working properly and of good quality, ideally the cable given with the enclosure.
The fact that it lists the GPU as present in lscpi and the connection as active is very promising, maybe try to see if there is an option to set a GPU as your primary GPU as I think what happens is that for battery reasons the AMD iGPU renders while a true GPU is not needed. Myabe launch a game and/or plugin the laptop to force it to switch? Idk, I haven’t had any iGPU in the past so I rely on what others told me, but if everything is detected thats a very good sign, maybe try searching internet for options in GNOME/KDE to manually select the main GPU. Maybe try apps for Nvidia in the flathub that allows to control things like this, it could be that the option to switch GPU is at an OS level more than a desktop/wayland level.
Also, (I only thought about this now) maybe your ubuntu problem of stutter is because you try to push wayyy to much data over that poor thunderbolt cable. What is the resolution of your screen? 4K, 2K, 1080p? I honestly don’t think any eGPU can output more than 4k 60fps or 1440p 120Hz realistically (might do 1080p 165Hz though). I mean you can set the output to that but then it will start to use horrible display stream compression settings, that might be why when setting it at 165Hz it bugs out on non-text items. That would be logical as non-text elements are more heavy to render as they cannot be completely kept in GPU memory so it is fetching data all the time, having to overcome the horrible display stream compression. That might be the root of the problem right here, eGPUs are not made for high fps gaming, cable tech is not here, even with the latest tech and USB 4 with 80Gb/s (very new standard of USB not present in framework laptops which are 40Gb/s I believe), I would highly doubt you could run 1440p 165 Hz. So maybe try downscaling to 1080p 165Hz or use 60Hz or 120Hz, test and see what works from what is available in FPS and resolutions for your screen. I assure you, 120Hz with no/few Display Stream Compression is ages ahead of 165Hz with heavy compression, both in term of image quality (negating the gains of 1440p) and responsiveness and latency (negating the gains of 165fps). Compression adds a TON of delays everywhere.
Tldr, nvidia GPUs really are a pain. I was so frustrated I sold my nvidia GPU and bought an AMD one. Drivers are also a pain in linux in general when they are not in the kernel, and GPU stuff is ALWAYS the drivers that has the most problems, because all distros do differently, because there’s so much versions (nvidia proprietary, nvidia first party “open source”, nvk, nouveau, nova, for AMD the proprietary drivers, the open source ones, and installing anything for Machine learning like OpenGL is a pain and always breaks…). Good luck mate.
. Problem like these could have so many causes between the OS, drivers, card, cables, frame, pcie link shielding…